Solution: Given: First Plane: $2 x-y+4 z=5$ [On multiply both the sides by $2.5]$ We obtain, $5 x-2.5 y+10 z=12.5 \ldots$ Second Plane: $5 x-2.5 y+10 z=6 \ldots$ Therefore, $\begin{array}{l}...
The planes: 2x – y + 4z = 5 and 5x – 2.5y + 10z = 6 are
Distance between the two planes: 2x + 3y + 4z = 4 and 4x + 6y + 8z = 12 is
A. 2 units
B. 4 units
C. 8 units
D. 2/√29 units
Solution: It is known to us that the distance between two parallel planes $A x+B y+C z=d_{1}$ and $A x+B y+C z=d_{2}$ is given as...
Prove that if a plane has the intercepts a, b, c and is at a distance of p units from the origin, then
Solution: It is known to us that the distance of the point $\left(\mathrm{x}_{1}, \mathrm{y}_{1}, \mathrm{z}_{1}\right)$ from the plane $\mathrm{Ax}+\mathrm{By}+\mathrm{Cz}$ $=\mathrm{D}$ is given...
Find the vector equation of the line passing through the point (1,2,-4) and perpendicular to the two lines: and
Solution: The vector eq. of a line passing through a point with position vector $\vec{a}$ and parallel to a vector $\overrightarrow{\mathrm{b}}$ is...
Find the vector equation of the line passing through and parallel to the planes and
Solution: The vector eq. of a line passing through a point with position vector $\vec{a}$ and parallel to a vector $\vec{b}$ is $\vec{r}=\vec{a}+\lambda \vec{b}$ It is given that the line passes...
Find the distance of the point from the point of intersection of the line and the plane
Solution: It is given that, The eq. of line is $\overrightarrow{\mathrm{r}}=(2 \hat{\mathrm{i}}-\hat{\mathrm{j}}+2 \hat{\mathrm{k}})+\lambda(3 \hat{\mathrm{i}}+4 \hat{\mathrm{j}}+2 \hat{\mathrm{k}})...
Find the equation of the plane which contains the line of intersection of the planes and And which is perpendicular to the plane
Solution: It is known that, The eq. of any plane through the line of intersection of the planes $\overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{n}_{1}}=\mathrm{d}_{1}$ and...
If O be the origin and the coordinates of P be (1, 2, –3), then find the equation of the plane passing through P and perpendicular to OP.
Solution: It is known to us that the eq. of a plane passing through $\left(\mathrm{x}_{1}, \mathrm{y}_{1}, \mathrm{z}_{1}\right)$ and perpendicular to a line with direction ratios $A, B, C$ is given...
Find the equation of the plane passing through the line of intersection of the planes and and parallel to -axis.
Solution: It is known to us that, The eq. of any plane through the line of intersection of the planes $\vec{r} \cdot \overrightarrow{n_{1}}=d_{1}$ and $\vec{r} \cdot \overrightarrow{n_{2}}=d_{2}$ is...
If the points (1, 1, p) and (–3, 0, 1) be equidistant from the plane , then find the value of .
Solution: It is known to us that the distance of a point with position vector $\vec{a}$ from the plane $\vec{r} \cdot \vec{n}=d$ is given as $\left|\frac{\vec{a} \cdot \vec{n}-d}{|\vec{n}|}\right|$...
Find the equation of the plane passing through the point (–1, 3, 2) and perpendicular to each of the planes x + 2y + 3z = 5 and 3x + 3y + z = 0.
Solution: It is known to us that the eq. of a plane passing through $\left(x_{1}, y_{1}, z_{1}\right)$ is given by $A\left(x-x_{1}\right)+B\left(y-y_{1}\right)+C\left(z-z_{1}\right)=0$ Where, A, B,...
Find the coordinates of the point where the line through (3, –4, –5) and (2, –3, 1) crosses the plane 2x + y + z = 7.
Solution: It is known to us that the eq. of a line passing through two points $A\left(x_{1}, y_{1}, z_{1}\right)$ and $B\left(x_{2}, y_{2}, z_{2}\right)$ is given as...
Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the ZX – plane.
Solution: It is known to us that the vector eq. of a line passing through two points with position vectors $\vec{a}$ and $b$ is given as...
Find the coordinates of the point where the line through (5, 1, 6) and (3, 4,1) crosses the YZ – plane.
Solution: It is known to us that the vector eq. of a line passing through two points with position vectors $\vec{a}$ and $\vec{b}$ is given as...
Find the shortest distance between lines
Solution: It is known to us that the shortest distance between lines with vector equations $\vec{r}=\overrightarrow{a_{1}}+\lambda \overrightarrow{b_{1}}$ and $\vec{r}=\overrightarrow{a_{2}}+\lambda...
Find the equation of the plane passing through (a, b, c) and parallel to the plane
Solution: The eq. of a plane passing through $\left(x_{1}, y_{1}, z_{1}\right)$ and perpendicular to a line with direction ratios $A, B, C$ is given as...
Find the vector equation of the line passing through (1, 2, 3) and perpendicular to the plane
Solution: The vector eq. of a line passing through a point with position vector $\vec{a}$ and parallel to vector $\vec{b}$ is given as...
If the lines are perpendicular, find the value of .
Solution: It is known to us that the two lines $\frac{x-1}{3 k}=\frac{y-2}{1}=\frac{z-3}{-5} \text { and }$ $\frac{\mathrm{x}-1}{3 \mathrm{k}}=\frac{\mathrm{y}-2}{1}=\frac{\mathrm{z}-3}{-5}$ are...
If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (–4, 3, –6) and (2, 9, 2) respectively, then find the angle between the lines AB and CD.
Solution: It is known to us that the angle between the lines with direction ratios $a_{1}, b_{1}, c_{1}$ and $a_{2}, b_{2}, c_{2}$ is given by $\cos \theta=\left|\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1}...
Find the equation of a line parallel to x – axis and passing through the origin.
Solution: It is known to us that, eq. of a line passing through $\left(x_{1}, y_{1}, z_{1}\right)$ and parallel to a line with direction ratios $a, b, c$ is...
Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b.
Solution: The angle between the lines with direction ratios $a_{1}, b_{1}, c_{1}$ and $a_{2}, b_{2}, c_{2}$ is given by $\cos \theta=\left|\frac{a_{1} a_{2}+b_{1} b_{2}+c_{1}...
If and are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are
Solution: Let's consider $l, m, n$ be the direction cosines of the line perpendicular to each of the given lines. Therefore, $ll_{1}+m m_{1}+n n_{1}=0 \ldots(1)$ And...
Show that the line joining the origin to the point (2, 1, 1) is perpendicular to the line determined by the points (3, 5, –1), (4, 3, –1).
Solution: Let's consider $O A$ be the line joining the origin $(0,0,0)$ and the point $A(2,1,1)$. And let $B C$ be the line joining the points $B(3,5,-1)$ and $C(4,3,-1)$ Therefore the direction...
In the following cases, find the distance of each of the given points from the corresponding given plane. Point Plane
(a) (2, 3, -5) x + 2y – 2z = 9
(b) (-6, 0, 0) 2x – 3y + 6z – 2 = 0
Solution: (a) The length of perpendicular from the point $(2,3,-5)$ on the plane $x+2 y-2 z=9 \Rightarrow x+2 y-2 z-9=0$ is $\frac{\left|a x_{1}+b y_{1}+c z_{1}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}}$...
In the following cases, find the distance of each of the given points from the corresponding given plane. Point Plane
(a) (0, 0, 0) 3x – 4y + 12 z = 3
(b) (3, -2, 1) 2x – y + 2z + 3 = 0
Solution: (a) The distance of the point $(0,0,0)$ from the plane $3 x-4 y+12=3 \Rightarrow$ $3 x-4 y+12 z-3=0$ is $\begin{array}{l} \frac{\left|a x_{1}+b y_{1}+c...
In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
(a) 4x + 8y + z – 8 = 0 and y + z – 4 = 0
Solution: (a) $4 x+8 y+z-8=0$ and $y+z-4=0$ It is given that The eq. of the given planes are $4 x+8 y+z-8=0 \text { and } y+z-4=0$ It is known to us that, two planes are $\perp$ if the direction...
In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
(a) 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0
(b) 2x – 2y + 4z + 5 = 0 and 3x – 3y + 6z – 1 = 0
Solution: (a) $2 x-2 y+4 z+5=0$ and $3 x-3 y+6 z-1=0$ It is given that The eq. of the given planes are $2 x-2 y+4 z+5=0$ and $x-2 y+5=0$ It is known to us that, two planes are $\perp$ if the...
In the following cases, determine whether the given planes are parallel or perpendicular, and in case they are neither, find the angles between them.
(a) 7x + 5y + 6z + 30 = 0 and 3x – y – 10z + 4 = 0
(b) 2x + y + 3z – 2 = 0 and x – 2y + 5 = 0
Solution: (a) $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$ It is given that The eq. of the given planes are $7 x+5 y+6 z+30=0$ and $3 x-y-10 z+4=0$ Two planes are $\perp$ if the direction ratio of the...
Find the angle between the planes whose vector equations are
Solution: It is given that The eq. of the given planes are $\vec{r}(2 \hat{i}+2 \hat{j}-3 \hat{k})=5 \text { and } \vec{r}(3 \hat{i}-3 \hat{j}+5 \hat{k})=5$ If $\mathrm{n}_{1}$ and $\mathrm{n}_{2}$...
Find the equation of the plane through the line of intersection of the planes x + y + z = 1 and 2x + 3y + 4z = 5 which is perpendicular to the plane x – y + z = 0.
Solution: Let's say that the eq. of the plane that passes through the two-given planes $x+y+z=1$ and $2 x+3 y+4 z=5$ is $\begin{array}{l} (x+y+z-1)+\lambda(2 x+3 y+4 z-5)=0 \\ (2 \lambda+1) x+(3...
Find the vector equation of the plane passing through the intersection of the planes and through the point
Solution: Let's consider the vector eq. of the plane passing through the intersection of the planes are $\overrightarrow{\mathrm{r}} \cdot(2 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}-3 \hat{\mathrm{k}})=7...
Find the equation of the plane through the intersection of the planes 3x – y + 2z – 4 = 0 and x + y + z – 2 = 0 and the point (2, 2, 1).
Solution: It is given that Eq. of the plane passes through the intersection of the plane is given by $(3 x-y+2 z-4)+\lambda(x+y+z-2)=0$ and the plane passes through the points $(2,2,1)$ Therefore,...
Find the equation of the plane with intercept 3 on the -axis and parallel to ZOX plane.
Solution: It is known to us that the equation of the plane $\mathrm{ZOX}$ is $\mathrm{y}=0$ Therefore, the equation of plane parallel to $\mathrm{ZOX}$ is of the form, $\mathrm{y}=\mathrm{a}$ As the...
Find the intercepts cut off by the plane 2x + y – z = 5.
Solution: It is given that The plane $2 x+y-z=5$ Let us express the equation of the plane in intercept form $x / a+y / b+z / c=1$ Where $a, b, c$ are the intercepts cut-off by the plane at $x, y$...
Find the equations of the planes that passes through three points.
(a) (1, 1, –1), (6, 4, –5), (–4, –2, 3)
(b) (1, 1, 0), (1, 2, 1), (–2, 2, –1)
Solution: (a) It is given that, The points are $(1,1,-1),(6,4,-5),(-4,-2,3)$. Let, $\begin{array}{l} =\left|\begin{array}{ccc} 1 & 1 & -1 \\ 6 & 4 & -5 \\ -4 & -2 & 3...
Find the vector and Cartesian equations of the planes
(a) that passes through the point and the normal to the plane is
(b) that passes through the point and the normal vector to the plane is
Solution: (a) That passes through the point $(1,0,-2)$ and the normal to the plane is $\hat{\mathrm{i}}+\hat{\mathrm{j}}-\hat{\mathrm{k}}$ Let's say that the position vector of the point $(1,0,-2)$...
In the following cases, find the coordinates of the foot of the perpendicular drawn from the origin.
(a) x + y + z = 1
(b) 5y + 8 = 0
Solution: (a) $x+y+z=1$ Let the coordinate of the foot of $\perp \mathrm{P}$ from the origin to the given plane be $P(x, y, z)$ $x+y+z=1$ The direction ratio are $(1,1,1)$ $\begin{array}{l}...
Find the Cartesian equation of the following planes:
(a)
Solution: Let $\overrightarrow{\mathrm{r}}$ be the position vector of $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ is given by $\overrightarrow{\mathrm{r}}=\mathrm{x} \hat{\mathrm{i}}+\mathrm{y}...
Find the Cartesian equation of the following planes:
(a)
(b)
Solution: (a) It is given that, The equation of the plane. Let $\overrightarrow{\mathrm{r}}$ be the position vector of $\mathrm{P}(\mathrm{x}, \mathrm{y}, \mathrm{z})$ is given by $\vec{r}=x...
Find the vector equation of a plane which is at a distance of 7 units from the origin and normal to the vector
Solution: It is given that, The vector $3 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-6 \hat{\mathrm{k}}$ Vector equation of the plane with position vector $\overrightarrow{\mathrm{r}}$ is $\vec{r} \cdot...
In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
(a) 2x + 3y – z = 5
(b) 5y + 8 = 0
Solution: (a) $2 x+3 y-z=5$ It is given that The eq. of the plane, $2 x+3 y-z=5 \ldots$. (1) The direction ratio of the normal $(2,3,-1)$ Using the formula,...
In each of the following cases, determine the direction cosines of the normal to the plane and the distance from the origin.
(a) z = 2
(b) x + y + z = 1
Solution: (a) $z=2$ It is given that The eq. of the plane, $z=2$ or $0 x+0 y+z=2 \ldots (1) .$ The direction ratio of the normal $(0,0,1)$ Using the formula, $\begin{array}{l}...
Find the shortest distance between the lines whose vector equations are
Solution: Consider the given equations $\begin{array}{l} \Rightarrow \vec{r}=(1-t) \hat{i}+(t-2) \hat{j}+(3-2 t) \hat{k} \\ \vec{r}=\hat{i}-t \hat{i}+t \hat{j}-2 \hat{j}+3 \hat{k}-2 t \hat{k} \\...
Find the shortest distance between the lines whose vector equations are and
Solution: It is known to us that shortest distance between two lines $\vec{r}=\overrightarrow{a_{1}}+\lambda \overrightarrow{b_{1}}$ and $\vec{r}=\overrightarrow{a_{2}}+\mu \overrightarrow{b_{2}}$...
Find the shortest distance between the lines and
Solution: It is known to us that the shortest distance between two lines $\frac{x+1}{7}=\frac{y+1}{-6}=\frac{z+1}{1}$ and $\frac{x-3}{1}=\frac{y-5}{-2}=\frac{z-7}{1}$ is given as:...
Find the shortest distance between the lines
Solution: It is known to us that the shortest distance between two lines $\overrightarrow{\mathrm{r}}=\overrightarrow{\mathrm{a}_{1}}+\lambda \overrightarrow{\mathrm{b}_{1}}$ and...
Show that the lines and are perpendicular to each other.
Solution: The equations of the given lines are $\frac{\mathrm{x}-5}{7}=\frac{\mathrm{y}+2}{-5}=\frac{\mathrm{z}}{1}$ and $\frac{\mathrm{x}}{1}=\frac{\mathrm{y}}{2}=\frac{\mathrm{z}}{3}$ Two lines...
Find the values of p so that the lines and are at right angles.
Solution: The standard form of a pair of Cartesian lines is:...
Find the angle between the following pairs of lines:
(i) and
(ii) and
Solution: Let's consider $\theta$ be the angle between the given lines. If $\theta$ is the acute angle between $\vec{r}=\overrightarrow{a_{1}}+\lambda \overrightarrow{b_{1}}$ and...
Find the vector and the Cartesian equations of the line that passes through the points (3, –2, –5), (3, –2, 6).
Solution: It is given that Let's calculate the vector form: The vector eq. of as line which passes through two points whose position vectors are $\vec{a}$ and $\vec{b}$ is...
Find the vector and the Cartesian equations of the lines that passes through the origin and (5, –2, 3).
Solution: Given: The origin $(0,0,0)$ and the point $(5,-2,3)$ It is known to us that The vector eq. of as line which passes through two points whose position vectors are $\vec{a}$ and $\vec{b}$ is...
The Cartesian equation of a line is Write its vector form.
Solution: It is given that The Cartesian equation is $\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2} \ldots \text { (1) }$ It is known to us that The Cartesian eq. of a line passing through a point...
Find the Cartesian equation of the line which passes through the point and parallel to the line given by
Solution: It is given that The points $(-2,4,-5)$ It is known that Now, the Cartesian equation of a line through a point $\left(\mathrm{x}_{1}, \mathrm{y}_{1}, \mathrm{z}_{1}\right)$ and having...
Find the equation of the line in vector and in Cartesian form that passes through the point with position vector and . is in the direction
Solution: Given: Vector equation of a line that passes through a given point whose position vector is $\vec{a}$ and parallel to a given vector $\vec{b}$ is $\vec{r}=\vec{a}+\lambda \vec{b}$ Let,...
Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector
Solution: Given that, Line passes through the point $(1,2,3)$ and is parallel to the vector. It is known to us that Vector eq. of a line that passes through a given point whose position vector is...
Show that the line through the points (4, 7, 8), (2, 3, 4) is parallel to the line through the points (–1, –2, 1), (1, 2, 5).
Solution: The points $(4,7,8),(2,3,4)$ and $(-1,-2,1),(1,2,5)$. Consider $A B$ be the line joining the points, $(4,7,8),(2,3,4)$ and $C D$ be the line through the points $(-1,-2$, 1), $(1,2,5)$. So...
Show that the line through the points (1, –1, 2), (3, 4, –2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).
Solution: Given that The points $(1,-1,2),(3,4,-2)$ and $(0,3,2),(3,5,6)$. Let's consider $A B$ be the line joining the points, $(1,-1,2)$ and $(3,4,-2)$, and $C D$ be the line through the points...
Show that the three lines with direction cosines Are mutually perpendicular.
Solution: Consider the direction cosines of $L_{1}, L_{2}$ and $L_{3}$ be $l_{1}, m_{1}, n_{1} ; l_{2}, m_{2}, n_{2}$ and $l_{3}, m_{3}, n_{3}$. It is known that If $\mathrm{f}_{1}, \mathrm{~m}_{1},...
Find the direction cosines of the sides of the triangle whose vertices are (3, 5, –4), (-1, 1, 2) and (–5, –5, –2).
Solution: Given that, The vertices are $(3,5,-4),(-1,1,2)$ and $(-5,-5,-2)$. Firstly find the direction ratios of $\mathrm{AB}$ Where, $A=(3,5,-4)$ and $B=(-1,1,2)$ Ratio of $A...
Show that the points (2, 3, 4), (–1, –2, 1), (5, 8, 7) are collinear.
Solution: If the direction ratios of two lines segments are proportional, then the lines are collinear. It is given that $\mathrm{A}(2,3,4), \mathrm{B}(-1,-2,1), \mathrm{C}(5,8,7)$ The direction...
If a line has the direction ratios –18, 12, –4, then what are its direction cosines?
Solution: Given that, The direction ratios are $-18,12,-4$ Where, $a=-18, b=12, c=-4$ Consider the direction ratios of the line as $\mathrm{a}, \mathrm{b}$ and $\mathrm{c}$ Direction cosines are...
Find the direction cosines of a line which makes equal angles with the coordinate axes.
Solution: Given that, Angles are equal. Let the angles be $\alpha, \beta, \mathrm{Y}$ The direction cosines of the line be I, $\mathrm{m}$ and $\mathrm{n}$ $I=\cos \alpha, m=\cos \beta \text { and }...
If a line makes angles 90°, 135°, 45° with the x, y and z-axes respectively, find its direction cosines.
Solution: Let's consider the direction cosines of the line be $I, m$ and $n$. Let $\alpha=90^{\circ}, \beta=135^{\circ}$ and $\mathrm{y}=45^{\circ}$ Therefore, $I=\cos \alpha, m=\cos \beta \text {...
If A and B be the points
and
, respectively, find the equation of the set of points P such that
, where k is a constant.
Given: The points A \[\left( \mathbf{3},\text{ }\mathbf{4},\text{ }\mathbf{5} \right)\]and B \[\left( \mathbf{1},\text{ }\mathbf{3},\text{ }\mathbf{7} \right)\]...
A point R with x-coordinate
lies on the line segment joining the points P
and Q
.Find the coordinates of the point R. [Hint Suppose R divides PQ in the ratio
. The coordinates of the point R are given by
Solution: Given: The coordinates of the points P \[\left( \mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{4} \right)\] and Q \[\left( \mathbf{8},\text{ }\mathbf{0},\text{ }\mathbf{10} \right)\]. Let...
Find the coordinates of a point on y-axis which are at a distance of
from the point P
.
Let the point on y-axis be A \[\left( 0,\text{ }y,\text{ }0 \right)\]. Then, it is given that the distance between the points A \[\left( 0,\text{ }y,\text{ }0 \right)\]and P \[\left( 3,-\text{...
. If the origin is the centroid of the triangle PQR with vertices P
, Q
and R
,then find the values of a, b and c.
Given: The vertices of the triangle are P (2a, 2, 6), Q (-4, 3b, -10) and R (8, 14, 2c). We know that the coordinates of the centroid of the triangle, whose vertices are \[({{x}_{1}},\text{...
Find the lengths of the medians of the triangle with vertices A
,B
and
.
According to the question: The vertices of the triangle are A \[\left( \mathbf{0},\text{ }\mathbf{0},\text{ }\mathbf{6} \right)\],B \[\left( \mathbf{0},\text{ }\mathbf{4},\text{ }\mathbf{0}...
Three vertices of a parallelogram ABCD are A
, B
and C
.Find the coordinates of the fourth vertex.
According to the question: ABCD is a parallelogram, with vertices A\[\left( \mathbf{3},\text{ }\text{ }\mathbf{1},\text{ }\mathbf{2} \right)\], B \[\left( \mathbf{1},\text{ }\mathbf{2},\text{...
Find the coordinates of the points which trisect the line segment joining the points P
and Q
.
Consider A \[({{x}_{1}},\text{ }{{y}_{1}},\text{ }{{z}_{1}})\]and B \[({{x}_{2}},\text{ }{{y}_{2}},\text{ }{{z}_{2}})\]trisect the line segment joining the points P \[\left( \mathbf{4},\text{...
Using section formula, show that the points A
, B
and C
are collinear.
Consider the point P divides AB in the ratio \[k:\text{ }1\]. By using section formula, So we have, Now, we check if for some value of k, the point coincides with the point C. Put \[\left( -k+2...
Find the ratio in which the YZ-plane divides the line segment formed by joining the points
and
.
Solution: Let the line segment formed by joining the points A \[\left(-\mathbf{2},\text{ }\mathbf{4},\text{ }\mathbf{7} \right)\]and B \[\left( \mathbf{3},-\text{ }\mathbf{5},\text{ }\mathbf{8}...
Given that A
, B
and C
are collinear. Find the ratio in which B divides AC.
Solution: Let us consider B divides AC in the ratio \[k:\text{ }1\]. By using section formula, So, we have \[9k\text{ }+\text{ }3\text{ }=\text{ }5\text{ }\left( k+1 \right)\] \[9k\text{ }+\text{...
Find the coordinates of the point which divides the line segment joining the points
and
in the ratio (i)
internally, (ii)
externally.
Solution: Let the line segment joining the points A \[\left( \text{ }\mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{5} \right)\]and B \[\left( \mathbf{1},\text{ }\text{ }\mathbf{4},\text{ }\mathbf{6}...
Find the equation of the set of points A, the sum of whose distances from p
and q
is equal to 10.
Let p \[\left( \mathbf{4},\text{ }\mathbf{0},\text{ }\mathbf{0} \right)\]& q \[\left( \text{ }\mathbf{4},\text{ }\mathbf{0},\text{ }\mathbf{0} \right)\] Let the coordinates of point A be (x, y,...
Find the equation of the set of points which are equidistant from the points
and
.
Solution: Let $A(1,2,3) \& B(3,2,-1)$ Let point $\mathrm{P}$ be $(\mathrm{x}, \mathrm{y}, \mathrm{z})$ Since it is given that po i.e. $P A=P B$ Firstly let us calculate Calculating PA $P...
Verify the following:
,
,
and
are the vertices of a parallelogram.
Let the points be: p\[\left( 1,\text{ }2,\text{ }1 \right)\], q\[\left( 1,\text{ }2,\text{ }5 \right)\],r\[\left( 4,\text{ }7,\text{ }8 \right)\] & s\[\left( 2,\text{ }3,\text{ }4 \right)\] pqrs...
Verify the following:
,
and
are the vertices of a right angled triangle.
Solution: Let the points be x\[\left( 0,\text{ }7,\text{ }10 \right)\], y\[\left( 1,\text{ }6,\text{ }6 \right)\]& z\[\left( \text{ }4,\text{ }9,\text{ }6 \right)\] Firstly let us calculate the...
Verify the following: (i) (0, 7, –10), (1, 6, – 6) and (4, 9, – 6) are the vertices of an isosceles triangle.
Solution: \[\left( \mathbf{0},\text{ }\mathbf{7},\text{- }\mathbf{10} \right)\] , \[\left( \mathbf{1},\text{ }\mathbf{6},\text{ }\text{ -}\mathbf{6} \right)\]and \[\left( \mathbf{4},\text{...
Show that the points
,
and
are collinear.
Solution: If three points are collinear, then they lie on a line. Firstly let us calculate distance between the 3 points i.e. AB, BC and AC Calculating AB A ≡ \[\left( \mathbf{2},\text{...
Find the distance between the following pairs of points:
and
Let P be \[\left( 2,\text{ }1,\text{ }3 \right)\]and y be \[\left( 2,\text{ }1,\text{ }3 \right)\] By using the formula, Distance Py = \[\sqrt{[{{({{x}_{2}}~\text{ }{{x}_{1}})}^{2}}~+\text{...
Find the distance between the following pairs of points:
and
Let x be \[\left( -1,\text{ }3,\text{ }\text{- }4 \right)\]and y be \[\left( 1,\text{ -}3,\text{ }4 \right)\] By using the formula, Distance xy = \[\sqrt{[{{({{x}_{2}}~\text{...
Find the distance between the following pairs of points:
and
Let x be \[(-3,7,2)\] and y be \[(2,4,-1)\] By using the formula, Distance xy = \[\sqrt{[{{({{x}_{2}}~\text{ }{{x}_{1}})}^{2}}~+\text{ }{{({{y}_{2}}~\text{ }{{y}_{1}})}^{2}}~+\text{...
Find the distance between the following pairs of points:
and
Given Let x be \[\left( \mathbf{2},\text{ }\mathbf{3},\text{ }\mathbf{5} \right)\]and y be \[\left( \mathbf{4},\text{ }\mathbf{3},\text{ }\mathbf{1} \right)\] Use the formula, Distance xy...
Fill in the blanks: (i) The x-axis and y-axis taken together determine a plane known as _______. (ii) The coordinates of points in the XY-plane are of the form _______. (iii) Coordinate planes divide the space into ______ octants.
Solution: (i) The x-axis and y-axis taken together determine a plane known as XY Plane. (ii) The coordinates of points in the XY-plane are of the form \[(x,y,0)\] (iii) Coordinate planes divide the...
Name the octants in which the following points lie:
,
,
,
,
,
,
,(2, – 4, –7).
Solution: The below table which represents the octants: (i) \[(1,2,3)\] Here x, y and z are positive. So it lies in I octant. (ii) \[(4,-2,3)\] Here x and z are positive, y is negative So it lies...
A point is in the XZ-plane. What can you say about its y-coordinate?
Solution: given a point is in XZ plane if a point is in XZ-plane then its y-co-ordinate is \[0\].
A point is on the x-axis. What are its y coordinate and z-coordinates?
Solution: Given a point is on the x-axis, if a point is on the x-axis, then the coordinates of y and z are \[0\]. Therefore, the point is \[(x,0,0)\]